Singular and nonsingular modules relative to a torsion theory

ABSTRACT

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.

KEYWORDS:

• τ-Essential submodule
• τ-semisimple module
• τ-singular module
• non-τ-singular module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S90
• 16D60
• 16D70

Acknowledgment

The authors would like to thank the referee for his/her valuable comments and suggestions which improved the presentation of this work.